a and b are sets too?
— litewave

No, but {a} and {b} are. — Banno

Well, in pure set theory a and b are sets too, because it's sets all the way down.

https://en.m.wikipedia.org/wiki/Set_theory#Ontology

So we can talk about a and b. And we can change the game a bit and talk about {a,b}. And in one way we have added a new thing to the conversation, yet in another way we are still just talking about a and b. We may be tempted to ask which way is real, but perhaps that question is irrelevant provided we talk clearly. — Banno

a and b are sets too?

Chairs are collections too.
— litewave
There's a whole new barrel of fish. — Banno

I already talked about that here:

Still, a set (collection) is also treated as a single object in set theory that exists as a single element in other sets. And I don't regard sets as "abstract" objects but rather as objects I can see all around me - there are sets of sets of sets etc. everywhere around. If a set is not an object in its own right then what objects are there? Just non-composite objects (like empty sets) at the bottom? And what if there is no bottom? One may object that there is no order of elements in a set while the sets we see around us are often ordered in intricate ways, but there are various ways of constructing ordered pairs out of unordered sets, for example the Kuratowski definition of an ordered pair. — litewave

I interact with collections of objects all the time.
— litewave
Sure. Just not int he way you interact with chairs. — Banno

Chairs are collections too.

What is a real metaphysical entity as contrasted with a real entity? What does the word "metaphysical" do here? — Banno

I was responding to your post in which you used the phrase "reified metaphysical entities". I understood them simply as real entities.

What does it mean to say they are real? What more can we do with real properties that we can't do just with properties? Or much better, with talk of sets or predicates? — Banno

It seems that we need real properties to explain in what ways things are similar to each other. The ways are the properties.

But I hope you see that your intuition - that having the property of being red and being a member of the set of red things say much the same thing - remains valid? — Banno

Yes.

That sets are objects in the ontology of set theory.
— litewave
And so long as you do not expect to bump in to them as you walk down the street, that's fine, isn't it? — Banno

I interact with collections of objects all the time.

Are you eliminating properties in favour of sets (which I would support), or making sets into reified metaphysical entities that ground properties? — Banno

I have always treated sets as real metaphysical entities. So if properties were sets, then properties would be real too. If properties are not sets, I am not sure if properties are real, but I tend to think they are.

But in set theory, sets do add to ontology.
— litewave
What does this mean? — Banno

That sets are objects in the ontology of set theory.

That is, you now have the set and the property, separately, and are apparently defining the set in terms of the property. — Banno

Yes, because my attempt to treat the set and the property as one and the same object seems to have failed.

Yes, so a model theorist? I — bongo fury

I have not studied model theory.

There’s no formal problem in set theory with counting sets as different from their elements. The “problem” arises only if one has an intuition that collections shouldn’t add to ontology—that a set should “just be” its members. In that case, the proliferation looks like an unnecessary or suspicious multiplication of entities. — Banno

But in set theory, sets do add to ontology. And in pure set theory all elements of a set are sets too.

Right, but, identify them with sets in the way that model theory maps predicates to sets? — bongo fury

I wanted to say that the set is the common property of its elements.

If we listen to Frank, then we have a, and we have b, of course; two things. But we also have the set {a,b}. So there are three things: a, b and {a,b}. But then we also have {a,b,{a,b}} - so there are four things in our domain - a, b, {a,b}, and {a,b,{ab}} - and off we go. I hope folk see the problem inherent in counting a set as a different thing to it's elements. — Banno

Why would there be a problem in counting a set as a different thing to its elements?

I don't know what pure set theory is. — Moliere

A set is pure if all of its members are sets, all members of its members are sets, and so on. For example, the set containing only the empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets.

https://en.m.wikipedia.org/wiki/Set_theory#Ontology

So how many individuals are there in the set {a,b}?

I say two. — Banno

Depends on what you mean by individual. There are obviously two elements in this set: a, b. By the way, in pure set theory these two elements are always sets as well.

As explained earlier, "identical" has a very specific definition here. The set is not extensionally identical to it's elements. — Banno

So the set is another object, in addition to its elements.

Glad to see you've since taken the vows of nominalism! — bongo fury

Alas, I have broken the vows in the course of this thread. Although it was not really nominalism about properties; I still regarded them as real separate objects, I just wanted to identify them with sets.

No, a set is no more than the things it contains. — Banno

Enter the prompt "Is a set identical to its elements?" in ChatGPT, Claude or Gemini. They will all give you the answer No.

To go back to the subset relation: if any element is a member of B, and B is a subset of A, then any element of B is an element of A. — Moliere

Right. There is a difference between "element" ("member") and "subset". Outside of set theory they may be both conflated with the concept of "part" but they are parts in different senses.

heh, well, once you flip your opinion I'll return -- but my work is done :D — Moliere

I see what you wrote early on:

But, no, they're different -- a set is its' elements, rather than a property which all the elements share. — Moliere

But I still don't think that a set is identical to its elements because a single object cannot be identical to multiple objects. So a set is another object, additional to its elements.

It seems that we have found genuinely different yet coextensive properties (like the property of redness and the property of being an instance of redness), so now I lean toward the view that properties cannot be identified with sets.

I would classify objects as concrete and abstract as you have indicated. The weirdness part would fall under abstract.

What I mean by "abstraction" is that you can treat the phone in either way without changing anything real. — Moliere

It depends on what we mean by "abstract" and "concrete". It is often said that concrete objects are located in space or in spacetime. Then a collection like a particular phone would be a concrete object. A collection consisting of a phone located in my house and another phone located in my friend's house would be a concrete object too, although some might resist that because the two phones are separated "too much". A collection consisting of my phone and of another phone in a different universe that is in a different spacetime might be regarded as a concrete object because its elements are located in a spacetime but then again, they are in different spacetimes, so this collection transcends a single spacetime. And then there is the general property/universal "phone" (or "phoneness") - that which all particular phones have in common - and I guess this would be regarded as an abstract object by almost anyone because unless we identify it with the set of all phones, it seems to transcend spacetime or be located in spacetime in an especially weird way.

I'd put it to you that the collection of individuals is an abstract object. To use your cell phone example -- we can think about the cell phone as a collection of particular objects and then name this in accord with set-theory. I.e. we can make sets which refer to concrete individuals, but to treat something as a set is still an abstraction. — Moliere

I'm not sure what you mean by "abstract" or "abstraction" here. Is the phone a concrete or an abstract object? Is it a collection of other objects or not?

Not in set theory. A set is criteria. It's an abstract object. — frank

Come on, objects that are included in a set satisfy certain criteria (have certain properties) but the set is a collection of those objects.

However, if "being an instance of redness" is referring to several things, as in "being an exemplification of redness", then it means the same as "redness". — RussellA

"Being an instance of redness" seems to be a property of all instances of redness, yet it seems to be a different property than redness itself. Both properties have exactly the same instances, which suggests that the properties could be one and the same, but "being an instance of redness" refers to an instance in relation to redness while redness refers only to redness.

You think knowledge is limited to what you can see? If so, she's never seen a set. A set is an abstract object. — frank

A set is a collection of objects. An average person surely knows what a collection is. Not so surely a universal.

She probably knows about redness as a universal. — frank

She has never seen a universal though. But she has seen collections (sets), so she may know more about collections than about universals.

If Karen doesn't know anything about sets, the substitution fails. — frank

She also doesn't know about the general property of redness, which probably cannot even be visualized. She only knows particular instances of redness.

So it makes sense to identify properties with sets. — RussellA

Or if not identify, then at least associate a set and a property like this:

set S = set of all elements that have property P

This is an intensional definition of a set, a definition by specifying a common property of the set's elements. An extensional definition of a set would be a definition by listing all the particular elements.

Are there really genuinely different coextensive properties? — RussellA

What about these two: the property of redness, and the property of being an instance of redness (or the property of having the property of redness). Both properties seem to be instantiated in all instances of redness, so the instances form one and the same set.

The set of all red things does not quite align with our intuitive grasp of “redness” as something unified and shared. A set is merely a collection of objects, whereas a property seems to be something more abstract—something that binds those objects together. — Astorre

Still, a set (collection) is also treated as a single object in set theory that exists as a single element in other sets. And I don't regard sets as "abstract" objects but rather as objects I can see all around me - there are sets of sets of sets etc. everywhere around. If a set is not an object in its own right then what objects are there? Just non-composite objects (like empty sets) at the bottom? And what if there is no bottom? One may object that there is no order of elements in a set while the sets we see around us are often ordered in intricate ways, but there are various ways of constructing ordered pairs out of unordered sets, for example the Kuratowski definition of an ordered pair.

So a set seems to be a single object that is something additional to its elements, not identical to any one of its elements, and not identical to multiple elements either, since it is a single object. A set somehow unifies/connects/binds its elements. In a sense, one could say that the elements "have" the set "in common", "share" it, or "participate" in it. So in this intuitive sense, a set seems evocative of a property and so I hoped to identify it with the common property of its elements, and thereby also get rid of property as a different kind of object and simplify the metaphysics of reality. But now it seems that there are genuinely different coextensive properties, which would dash the hope of identifying properties with sets.

Admittedly there is also something about the concept of a set that seems a bit jarring with the concept of a common property of the set's elements: a set encompasses or aggregates the elements with both their common and different properties, while a common property of the elements seems to be some commonality that is as if distilled/extracted from the elements rather than the result of encompassing or aggregating the elements. This may be what felt misaligned to you too.

Identifying properties such as “equilateral triangle” and “equiangular triangle” as one and the same disregards their contextual distinctions. In geometric analysis, for example, whether emphasis is placed on sides or angles can carry significant implications, even if the extension is the same. — Astorre

The properties “equilateral triangle” and “equiangular triangle” don't seem meaningfully different to me in any way. One description mentions the equality of sides and the other the equality of angles but the concept of triangle includes both sides and angles and is such that the equality of sides logically necessitates the equality of angles, and vice versa. What implications would the different emphasis in description have for geometrical analysis?

In the end, your approach requires a metaphysical commitment to the reality of possible worlds, which is itself a contested position. — Astorre

I lean to modal realism because I don't see a difference between the existence of a possible (logically consistent) object as "real" and as "merely possible". Logical consistency seems to be just existence in the broadest sense. The challenge is to find which objects are logically consistent, because they must be consistent with everything else in reality (like in mathematics - everything either fits together or falls apart). This might involve looking for the necessary properties or relations of any possible object or analysis of the concept of "object" or "something" itself and build from that. But if reality is complex enough to include the set of natural numbers (arithmetic) then it is impossible to prove that our description of it is consistent, as per Godel's second incompleteness theorem. Sensory detection of objects helps us find consistent objects but the senses have their limitations too.

I propose we step away from a substantialist approach to ontology and turn instead toward a processual one, which I am actively developing. — Astorre

I imagine sets as the fundamental objects in reality, from which everything else might be explained (properties as general objects could be fundamental too, if they are consistent objects other than sets). I am no set theorist or mathematician but my methaphysics is strongly influenced by the wide acceptance of pure set theory as a foundation of mathematics, in which all mathematical concepts can in principle be expressed as pure sets. It would explain the mathematical aspect of reality, if reality consists of sets. A space can be defined as a set with a continuity between the sets inside this set (as defined in general topology). Time can be defined as a special kind of space, as defined in theory of relativity. So it seems that a spacetime, with its spatiotemporal, including causal, relations, could be a certain kind of set. But then spacetimes would be just certain parts of a much greater reality where all possible (logically consistent) sets exist.

Having the property red is not the same as the property red. — frank

Hm yes, the problem will be in the property of being red, which I equated with these two properties. It seems ok to equate being red with having the property red. But being red should not be equated with redness when redness is meant as general redness while being red is meant as particular redness.

Anyway, if there are genuinely different properties that have the same set of instances (for example, properties like general redness and particular redness), then my OP proposal of identifying a property with the set of its instances fails.

Membership in the red set entails having red as a property. Entailment doesn't get you to identity, though. Or if so, how? — frank

This way:

For example, let's take property red or redness (X = red): The property of "being in set red" is the same as the property of "having property red", which is the same as the property of "being red", which is the same as property red. So, the property of "being in set red" and property red are one and the same property. — litewave

So if I say the peony is red, I mean it's in the set of all red things. So did we change from the set is the property to being in the set is the property? — frank

These two properties have exactly the same instances and if I got it right, they are one and the same property, just described differently.

This is interesting but confusing. Is "Being in that set means having that property" different from "'Being in that set' is a property of the pebble"? I thought we didn't want set membership to count as a property. — J

On further thought, I find this confusing too. The property of "being in set X" may seem to be the property of members of set X, but perhaps it is actually the property of the set membership relation instead. (A member is in set X but "being in set X" is not the property of the member but of the set membership relation between the member and set X.) Similarly, the property of "having property X" may seem to be the property of instances of property X, but perhaps it is actually the property of the instantiation relation instead. (An instance has property X but "having property X" is not the property of the instance but of the instantiation relation between the instance and property X.)

Since I equate set with property, members of set are equated with instances of property, and set membership relation is equated with instantiation relation. — litewave

My reply above was a groaner, wasn't it. Perhaps the property of "being in set X" could be interpreted as a property of the set membership relation but it is clearly a property of elements of set X, first and foremost.

So, given that I propose identifying property X with the set of instances of property X, it seems that the elements of set X share two properties: property X and the property of "being in set X". And these two properties have the same extension - all elements of set X, so they are coextensive properties. However, I think that these two properties are not really different; they are one and the same property, just described differently. The property of "being in set X" is the same as the property of "having property X", which is the same as the property of "being X", which is the same as property X. So, the property of "being in set X" and property X are one and the same property.

For example, let's take property red or redness (X = red): The property of "being in set red" is the same as the property of "having property red", which is the same as the property of "being red", which is the same as property red. So, the property of "being in set red" and property red are one and the same property.

Can we agree that only one possible world actually exists (the actual world)?

In that case, your set includes "things" that do not exist, never have existed, and never will exist (they are non-actual possibilities). Let's focus on this subset of your big set. Does it have any members? Are the members things? If so, what is a thing? — Relativist

If only the actual world exists, then a property has instances only in the actual world, and the property is still a set of its instances (but the instances exist only in the actual world). The instances of a property are whatever has the property.

This is interesting but confusing. Is "Being in that set means having that property" different from "'Being in that set' is a property of the pebble"? I thought we didn't want set membership to count as a property. — J

On further thought, I find this confusing too. The property of "being in set X" may seem to be the property of members of set X, but perhaps it is actually the property of the set membership relation instead. (A member is in set X but "being in set X" is not the property of the member but of the set membership relation between the member and set X.) Similarly, the property of "having property X" may seem to be the property of instances of property X, but perhaps it is actually the property of the instantiation relation instead. (An instance has property X but "having property X" is not the property of the instance but of the instantiation relation between the instance and property X.)

Since I equate set with property, members of set are equated with instances of property, and set membership relation is equated with instantiation relation.

So you're saying that having a property is a matter of being a member of the set of all things that have that property. That's trivially true. — frank

The point of my OP is that the set actually is the property. That may not be obvious.

So the peony has the set of all red things. How does it have that set? — frank

By being an element of the set, thus having what all the other elements of the set have.

. The property of redness is the set of all red things.
2. A peony has the property of redness.
3. A peony has the set of all red things.

Help me out here. That doesn't make sense. — frank

It sounds weird if when you think of the set you think of all the red things. It makes you think that the peony somehow has all the red things, which is absurd. But the set is not all the red things. It is something else, which all the red things have in common.

So in a way what I'm asking here is to say "How does this notion of unification fit within a strict logical definition?" — Moliere

Well, in predicate logic you have individuals that have/satisfy a property/predicate. I propose that the property is the set of these individuals.